Adaptive Networks

Ties that rewrite the nodes that rewrite the ties. Agreement can thicken a link; discord can strain and break it; exploration can open a new bridge. Consensus, echo chambers and fragmentation are not preset layouts—they are histories the network remembers in its changing topology.

A conventional network model separates two questions. Dynamics on networks asks how a process—opinion change, infection, synchronisation or diffusion—unfolds over a fixed graph. Dynamics of networks asks how edges and nodes are added or removed. An adaptive network combines them: the process changes the graph, and the changed graph alters the process.

Gross and Blasius summarised the defining feature as coevolution between network topology and node state. This feedback appears in many domains. Susceptible individuals can avoid infectious neighbours by rewiring contacts; collaborators can leave unproductive partnerships; social ties can form through similarity; ecological interactions can change as populations adapt. The same mathematical architecture can produce very different interpretations.

In AL.Floys, every node has a continuous state s in [0,1]. The engine does not assign a literal meaning such as political ideology, belief truth, identity or emotion. State colour is an abstract scalar. An edge represents a relationship through which influence may occur, and in the rich mode it also carries weight, compatibility, memory, strain and birth time.

The initial graph is Erdős–Rényi-like: random node pairs are connected until a target edge count corresponding to the requested mean degree is reached. Rich-mode states begin uniformly across [0,1]; Fragmentation Lab begins with two broad state bands. A selected fraction of nodes receives a stubborn flag.

The default Relational Field pack combines bounded-confidence influence with adaptive weighted edges.

For an edge between nodes a and b, the state gap is

d = |s_a − s_b|.

A global confidence radius is derived from Openness. Influence is accepted softly rather than with a hard all-or-nothing threshold:

acceptance = max(0, 1 − d / radius).

Connected nodes then move symmetrically toward one another at a rate controlled by Influence, multiplied by edge weight and acceptance. The code computes all pulls from the pre-event states and accumulates them before updating, avoiding arbitrary edge-order bias within that tick.

This is related to bounded-confidence opinion models such as Deffuant–Weisbuch, where agents interact only when their difference is below a confidence bound. It is not an exact implementation of that pairwise asynchronous model. AL.Floys uses a weighted network, simultaneous edge influence and soft linear acceptance.

After state influence, the edge itself adapts. Agreement raises a target weight and compatibility. Memory accumulates according to repeated agreement. Discord increases strain. High strain combined with low weight creates a stochastic break hazard. Newly formed edges receive a grace period before they can break.

When a tie breaks, one endpoint seeks a replacement. Candidate selection mixes several strategies: friends-of-friends, state-similar nodes, state-dissimilar nodes and random non-neighbours. Curiosity increases the number of exploration attempts; Loyalty slows plasticity, reduces break hazard and extends grace.

This mechanism is not “the network finds the optimal structure.” It is local path-dependent adaptation. The resulting graph can preserve connected diversity, collapse toward consensus, form segregated modules or retain fragile cross-community bridges.

Drift adds independent state perturbation. Without drift, a sufficiently aligned network can freeze. With drift, small differences continually re-enter and relationships must keep adapting. Noise sustains activity but does not itself choose a socially desirable outcome.

The second rule pack, Fragmentation Lab, is described in code as a canonical adaptive-voter reduction. The broad lineage is correct, but the implementation should be called adaptive-voter-inspired rather than canonical.

At each model tick, it collects edges whose endpoint states differ beyond a discord threshold. One discordant edge is sampled. With probability p_rewire, the edge is removed and one endpoint reconnects to a random non-neighbour while total edge count is conserved. Otherwise, non-stubborn endpoints move partially toward one another. Drift can then perturb non-stubborn states.

Classic adaptive-voter models often use discrete binary states and one-node adoption. AL.Floys uses continuous states, potentially moves both endpoints toward each other and selects a random replacement rather than necessarily rewiring to the same state. It captures the competition between adoption and rewiring and can exhibit connected versus fragmented regimes, but it is not numerically identical to one canonical paper.

A central phase-transition idea in adaptive-voter research is that high rewiring can remove discordant edges faster than state influence resolves them. The graph then fragments into internally homogeneous components. Low rewiring keeps incompatible nodes connected long enough for state adoption to produce consensus. The important variable is the relative rate of state change versus topology change.

AL.Floys adds a structural-analysis layer. It computes connected components, largest-component fraction, state variance, state entropy, discordant-edge fraction, cross-community links and cumulative topology turnover. A transparent rules-based classifier with hysteresis labels the current regime as Consensus, Connected diversity, Fragmenting or Transitional.

Those labels are useful editorial instruments, not discovered natural kinds. The thresholds are authored: for example, Consensus requires a large connected component, low state variance and few discordant edges. The classifier should be described as a diagnostic heuristic, not an objective theorem about social systems.

The engine also identifies bridges in the graph-theoretic sense: edges whose removal would increase the number of connected components. These are found with a standard bridge algorithm and highlighted. A visually thin bridge can be structurally decisive; a thick edge inside a dense community may be redundant.

Communities are inferred through deterministic weighted label propagation. The resulting labels drive coloured territories and a small layout regularisation, but the code explicitly states that community detection must never feed back into scientific dynamics. The visible territorial regions are analysis overlays.

This distinction extends to the entire spatial layout. Nodes do not inhabit a physical plane used by the model rules. The force-directed layout places connected nodes together, repels unconnected nodes locally, applies a weak community pull and blends each node 20% toward an angle determined by its state. Screen distance is therefore an explanatory lens. Influence travels along edges, not because two circles happen to be close on the canvas.

A visitor can easily misread the animation as spatial social segregation. The report should say clearly:

The graph is the model; the geometry is a changing diagram of the graph.

P0 control defect: Openness is inverted in Fragmentation Lab

The public Openness help says “how much difference a relation can hold before influence stops.” In Relational Field, higher Openness correctly increases the confidence radius.

In Fragmentation Lab, however, the discord threshold is calculated as

threshold = 0.55 − 0.35 × openness.

Higher Openness therefore makes smaller differences count as discordant. The temporary Open intervention subtracts still more from the threshold, making the selected node less tolerant rather than more tolerant. This is the opposite of the public semantics.

The mapping must be reversed and all fragmentation presets rebenchmarked. An exact replacement formula should be chosen through calibration, but the monotonic contract is non-negotiable: more openness must allow more difference before an edge is treated as discordant.

P0 control defect: Loyalty and Curiosity are inert in Fragmentation Lab

The public simple surface continues to expose Loyalty and Curiosity after a Fragmentation preset loads. The fragmentation coefficients use only Openness, Influence, Drift and Rewire probability. Loyalty and Curiosity have no effect.

Two options are acceptable:

1. hide or disable the controls in Fragmentation Lab and explain its reduced parameter set; or
2. define mode-specific meanings and implement them.

The first is cleaner. A research guide cannot ask the visitor to infer causation from no-op controls.

P1 semantic defect: zealots are only stubborn in one mode

The advanced Zealot fraction help says stubborn nodes never adopt. Fragmentation Lab honours that rule. Relational Field does not check the stubborn flag during symmetric influence or drift, so those nodes change like ordinary nodes. The flag only prevents direct Persuade interventions.

Either hide Zealot fraction outside Fragmentation Lab or implement a mode-appropriate stubbornness rule. Until then, presets and copy must not imply that Relational Field contains immutable zealots.

P1 visual–causal mismatch: node openness membranes show unused heterogeneity

Every node receives an individual base openness value, and the renderer changes its outer membrane according to base openness plus temporary boost. Yet both science rule packs use global Openness and the temporary boost; they do not use the node’s base openness in acceptance.

The viewer therefore sees heterogeneous “open” and “closed” membranes that are not causing heterogeneous influence tolerance. Either incorporate node-level openness into the rules or render only the global parameter and active temporary boost.

P1 interaction semantics: Persuade always moderates toward 0.5

The Persuade/Nudge tool has no directional target. Clicking a non-stubborn node moves its state toward the midpoint 0.5. It does not persuade toward the cursor, the user’s selected position on a state axis or another node.

The present tool should be renamed Moderate or Centre. A richer Persuade tool could first select a source node and then a target, or map drag direction to a desired state.

Random-graph theory supplied the fixed-network background, including Erdős and Rényi’s foundational models of random connectivity. Opinion and interaction models then studied how node states evolve over those graphs.

Deffuant, Neau, Amblard and Weisbuch’s 2000 bounded-confidence model made tolerance explicit: sufficiently similar agents move toward one another, while agents beyond the confidence threshold do not interact. This offered a minimal route to consensus, polarisation and persistent clusters of opinion.

In 2006, Petter Holme and M. E. J. Newman combined opinion convergence with network formation between similar agents and found a nonequilibrium transition between diverse and largely aligned regimes. The same year, Gross, D’Lima and Blasius showed how behavioural rewiring in an epidemic network could produce correlations, oscillations and hysteresis.

Gross and Blasius’s 2008 review consolidated these studies under adaptive coevolutionary networks: systems where node dynamics and topology change together through local rules.

Vázquez, Eguíluz and San Miguel’s 2008 adaptive-voter analysis made fragmentation especially clear. Discordant edges either rewire or trigger adoption, and the system crosses between connected and fragmented absorbing phases depending on the balance.

AL.Floys deliberately combines several ideas from this lineage rather than reproducing one paper. Relational Field is a custom weighted, memory-bearing and continuously valued adaptive network. Fragmentation Lab is a reduced rewire-or-adopt system designed to expose the canonical competition more directly.

Aspect	Current implementation
Nodes	Active graph nodes with stable IDs, continuous state in [0,1], a stubborn flag, base openness and display position.
Edges	Undirected adaptive ties with weight, compatibility, memory, strain and birth tick in Relational Field.
Initial graph	Erdős–Rényi-like simple graph with exact target edge count from Node count and Mean degree.
Relational Field influence	Symmetric weighted soft bounded-confidence pull from pre-tick states, followed by independent drift.
Relational Field topology	Agreement/discord adapt edge properties; strained weak ties break stochastically and one endpoint rewires through a mixed search strategy.
Fragmentation Lab	Sample one discordant edge; rewire with probability p or move non-stubborn endpoints toward each other; edge count conserved.
Clock	Fixed 30 Hz model microsteps, runtime dt clamped and accumulated.
Interventions	Bridge, Sever, Persuade/centre and temporary Open boost.
Diagnostics	Components, largest component, state variance/entropy, discord, cross-community links, turnover, graph bridges and heuristic regime classification.
Community analysis	Deterministic weighted label propagation used for rendering and diagnostics, not science dynamics.
Layout	Force-directed explanatory lens with springs, repulsion, community regularisation and a state-angle bias.
Rendering	State colour, degree-scaled soft nodes, openness membrane, strain-thinned fibres, bridge glow, territories and event pulses.
Public simple controls	Openness, Influence, Loyalty, Curiosity, Drift and Nodes.
Public tools	Bridge, Sever, Persuade/Nudge and Open.
Known defects	Fragmentation Openness is inverted; Loyalty/Curiosity are no-ops there; zealot semantics and node-openness rendering are mode-inconsistent.

Consensus — alignment without disappearance of the graph

Node colours converge while the largest component remains connected. Consensus is a state pattern, not simply one dense visual cluster.

Connected diversity — difference persists across cross-links

Several state regions remain distinct while bridge edges keep most nodes in one component. The network contains diversity without complete segregation.

Echo chambers — state sorting becomes topology

Discordant ties weaken and break; rewiring creates modules with internally similar states and fewer cross-links. Preference has become structural separation.

Bridge fragility — one edge carries disproportionate connectivity

Highlighted bridge ties are graph-theoretically load-bearing. Severing one can split the network even if it appears visually modest.

Topology turnover — relationships remember and rewrite history

Edges strengthen, weaken, disappear and reform. The final graph depends on the path of earlier interactions and interventions.

Fragmentation threshold — rewiring outruns influence

In Fragmentation Lab, high rewire probability can remove discordant contact before states converge, freezing separate components.

Stubborn influence — connectivity determines reach

A stubborn node can only affect nodes connected through the evolving graph. A zealot’s numerical state alone does not determine its macroscopic impact.

Layout versus graph — apparent distance can mislead

Two nodes may drift close on screen without an edge and therefore exert no state influence. Conversely, a long visible edge remains a direct relationship.

First 30 seconds

1. Load Open Federation and identify coloured communities and highlighted bridge ties.
2. Ignore physical distance and follow one edge: watch its thickness or strain change as endpoint states evolve.
3. Use Bridge to connect two visibly distinct communities. Ask whether the new edge remains, changes state flow or becomes strained.
4. Load Echo Chambers and compare the largest connected component with Open Federation.
5. Use Sever on a highlighted bridge. Watch whether one intervention changes the connected-component regime.

Do not use Openness as a Fragmentation Lab experiment until its sign is fixed, and do not use Loyalty or Curiosity there until they are hidden or implemented.

Three experiments

Experiment	Question	Do	Watch for	Why it matters
Bridge versus similarity	Can one new relationship keep different communities connected?	Open Federation or One Bridge → add one cross-community Bridge.	State influence, tie strain, survival or later break.	Shows that topology can preserve contact without guaranteeing consensus.
Rewire-or-adopt competition	When does discord produce agreement rather than separation?	Fragmentation Lab → compare low and high Rewire probability, holding Influence fixed.	Connected convergence versus frozen components.	Exposes the canonical adaptive-network competition.
Structural vulnerability	Is every edge equally important?	One Bridge → sever a highlighted bridge, then sever a redundant internal edge.	Component split versus little connectivity change.	Distinguishes edge weight from graph-theoretic bridge status.

A fourth advanced experiment is to load Counterfactual Echo, capture a state and run two branches—one with a bridge intervention and one with a sever—to compare path-dependent outcomes from the same starting network.

Parameter	What it really controls	Increase it	Decrease it	What to watch
Openness	Relational Field: width of soft bounded confidence. Fragmentation: currently inverted and defective.	Rich mode accepts larger state differences.	Rich mode restricts influence to similar states.	Fix Fragmentation mapping before cross-mode comparison.
Influence	Rate at which accepted connected nodes move toward one another.	Faster state convergence across viable edges.	Slower state change relative to topology.	Whether ties break before they can reconcile states.
Loyalty	Rich mode: slower edge plasticity, longer grace and lower break hazard.	More persistent ties and relational memory.	Faster strain-driven breakage.	No effect in Fragmentation Lab under current code.
Curiosity	Rich mode: number of attempts to explore replacement partners.	More active rewiring search.	More limited local replacement.	No effect in Fragmentation Lab under current code.
Drift	Independent state noise.	Sustained disagreement and ongoing adaptation.	Easier freezing into consensus or stable clusters.	State variance and topology turnover.
Nodes	Population size; structural.	Larger network and more possible communities.	Smaller, easier-to-follow graph.	Finite-size sensitivity and bridge frequency.
Mean degree (advanced)	Initial average number of relationships per node.	More redundant paths and fewer structural bridges.	Sparser graph and greater fragility.	Largest component and bridge count.
Rewire probability (Fragmentation)	Probability of topology change rather than state adoption on a discordant edge.	Greater fragmentation tendency.	Greater consensus tendency.	Connected versus fragmented absorbing states.
Zealot fraction (advanced)	Fraction flagged stubborn; currently only scientifically effective in Fragmentation Lab.	More non-adopting nodes in that mode.	Fewer fixed-state anchors.	Hide or repair in Relational Field.

Open Federation

The reference connected-diversity regime. Persistent communities remain linked by cross-ties. It is the best first world for Bridge interventions.

Echo Chambers

Low exploration and weak tie persistence promote modular segregation. Use it to study how state preference becomes structural disconnection.

Critical Edge

A Fragmentation Lab regime near a transition where one intervention can alter the final connectedness. Avoid changing Openness until the inversion bug is fixed.

One Bridge

Communities are connected through very few load-bearing ties. It is the strongest preset for comparing structural bridges with ordinary edges.

Bridge Builders

High curiosity creates replacement ties that preserve a large connected component despite community structure. It demonstrates adaptive connectivity rather than inevitable homophily.

The Zealot

Fragmentation mode with a larger stubborn fraction. Use it to ask whether stubbornness matters more through state rigidity or through where the stubborn nodes sit in the graph.

Memory Trap

High loyalty preserves local structure. The preset exposes path dependence: relationships can survive because their accumulated memory lowers effective strain.

Aligned Hubs

Denser connectivity and stronger influence pull states toward consensus. It is useful for showing how network redundancy and state dynamics reinforce one another.

Counterfactual Echo

A near-threshold fragmentation regime designed for snapshot branching. It is the best preset for exact same-start intervention comparisons.

Recommended learning order: Open Federation → One Bridge → Echo Chambers → Bridge Builders → Aligned Hubs → Critical Edge → The Zealot → Memory Trap → Counterfactual Echo.

• Bridge creates a new edge with moderate initial weight, compatibility and memory. It changes the scientific topology directly.
• Sever removes a selected edge. Its consequence depends on graph structure: cutting a bridge can split a component; cutting a redundant edge may not.
• Persuade / Nudge currently moves a selected non-stubborn node toward state 0.5. Rename Moderate unless a directional target is implemented.
• Open temporarily widens acceptance in Relational Field for roughly ninety model ticks. In Fragmentation Lab the current sign is reversed and must be fixed before the tool is scientifically interpretable.

Unlike generic Force tools, these interventions operate on explicit graph semantics. That makes Adaptive Networks especially suitable for counterfactual experiments with captured states.

Nodes are not people, communities, political parties or biological species unless an application supplies that mapping. State 0 and state 1 are abstract endpoints, not moral or factual opposites. “Consensus” is a low-variance diagnostic, not proof of truth, justice or social welfare.

Adaptive does not mean intelligent, optimal or beneficial. Rewiring can maintain diversity, amplify segregation or destroy connectivity. The model has no strategic reasoning, utility, language, institutional power or empirical social calibration.

The force layout is not geographic or social distance. Community territories are analytical overlays. State-angle layout bias intentionally makes opinion difference visible; it is not a causal field.

Relational Field is a custom synthesis, not a canonical published equation. Fragmentation Lab is adaptive-voter-inspired rather than an exact binary voter process. Regime labels use authored thresholds and hysteresis.

Current mode defects materially limit public interpretation: Openness is reversed in Fragmentation Lab, Loyalty and Curiosity are inert there, and zealot semantics differ by mode.

Adaptive Networks contributes co-evolution of structure and state. Flock and Swarmalators change agent motion while interaction rules remain structurally similar. Stigmergy stores influence in an environmental field. Adaptive Networks makes the relationship graph itself historical and plastic. It shows that a system’s architecture is not always the stage on which dynamics occurs; architecture can be one of the dynamical variables.

Foundational and primary sources

1. Guillaume Deffuant, David Neau, Frédéric Amblard and Gérard Weisbuch, “Mixing beliefs among interacting agents,” Advances in Complex Systems 3, 87–98 (2000). DOI: 10.1142/S0219525900000078.
2. Petter Holme and M. E. J. Newman, “Nonequilibrium phase transition in the coevolution of networks and opinions,” Physical Review E 74, 056108 (2006). DOI: 10.1103/PhysRevE.74.056108.
3. Thilo Gross, Carlos J. Dommar D’Lima and Bernd Blasius, “Epidemic dynamics on an adaptive network,” Physical Review Letters 96, 208701 (2006). DOI: 10.1103/PhysRevLett.96.208701.
4. Federico Vázquez, Víctor M. Eguíluz and Maxi San Miguel, “Generic Absorbing Transition in Coevolution Dynamics,” Physical Review Letters 100, 108702 (2008). DOI: 10.1103/PhysRevLett.100.108702.

Modern context

5. Thilo Gross and Bernd Blasius, “Adaptive coevolutionary networks: a review,” Journal of the Royal Society Interface 5, 259–271 (2008). DOI: 10.1098/rsif.2007.1229.