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A degree distribution assigns the fraction or probability of vertices having each degree of a vertex. For an observed network, $P(k)=\#\{i:k_i=k\}/n$. Preserving every labelled degree preserves this distribution, but preserving only its mean does not.

For the Barabási-Albert model, choose a connected seed graph on $m_0$ vertices, the number $m\le m_0$ of new edges per added vertex, and a final size $n$. A complete seed with $m_0\ge 2$ suffices. Add one vertex at a time, attaching it to $m$ distinct existing vertices selected with weights proportional to their current vertex degrees. Exclude duplicate edges and self-loops.

Increasing $n$ creates a larger network and permits older hubs to grow. Increasing $m$ makes the graph denser, with asymptotic average vertex degree $2m$, and tends to shorten graph distances. The seed affects early structure and the oldest hubs; standard linear preferential attachment gives a robust large-size tail exponent.

Its asymptotic degree distribution is $P(k)\propto k^{-3}$, so it is a scale-free network. Hubs produce short average shortest path length, but elevated clustering coefficients are not automatic: $m=1$ with a tree seed gives a triangle-free tree, and clustering decreases with size in the standard model. The strict clustered small-world network property is therefore not guaranteed.^[ref]

For the Watts-Strogatz model, choose $n$ vertices on a ring, an even initial vertex degree $K<n$, and a rewiring probability $p$. Connect each vertex to its $K/2$ nearest neighbours on each side. Consider each original clockwise edge once: with probability $p$, retain its starting endpoint and replace its other endpoint uniformly, excluding self-loops and duplicate edges. The edge count remains $nK/2$, although individual vertex degrees change.

Increasing $n$ at fixed $K$ lengthens unrewired-ring graph distances. Increasing $K$ adds edges and shortens distances. Increasing $p$ introduces shortcuts, sharply reducing average shortest path length at first, then progressively reducing the clustering coefficient by destroying local triangles.

The small-world network regime occurs at intermediate $p$ with sufficient shortcuts, substantial clustering, and $K\ll n$. At $p=0$ distances are long; near $p=1$ the clustering advantage disappears. The degree distribution remains concentrated around $K$, so this is not a scale-free network.^[ref]

For the Erdős-Rényi model $G(n,p)$, choose the number $n$ of vertices and an edge probability $0\le p\le 1$. Visit every unordered pair of distinct vertices once and include its edge independently with probability $p$. This construction gives a simple graph with expected edge count $p\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ and mean vertex degree $p(n-1)$.

Increasing $p$ at fixed $n$ makes the graph denser, increases typical vertex degree and local clustering coefficient, and tends to shorten graph distances and improve connectivity. Increasing $n$ at fixed $p$ also increases the mean vertex degree; to retain a sparse comparison one instead decreases $p$ approximately as $1/n$.

Its degree distribution is binomial distribution, approximately Poisson distribution in the sparse large-$n$ regime, so it is not a scale-free network. Connected realizations can have short average shortest path length, but their clustering coefficients are not elevated above the random baseline they themselves define. Thus they are not small-world networks under the high-clustering convention, although “small world” is sometimes used more loosely to mean only short distances.

The supplied adjacency matrix describes a four-vertex cycle graph, with every vertex degree equal to two. Degree-preserving edge rewiring is random, so it does not determine one unique labelled output adjacency matrix. For example, replace edges $AB$ and $CD$ by $AC$ and $BD$, retaining $AD$ and $BC$. This valid switch gives, in order $A,B,C,D$,Each row still sums to two. Under the simple graph convention, all allowed outputs have the same unlabelled four-cycle topology: there are three possible labelled four-cycles. A rejected switch or later return can also leave the original adjacency matrix.

It is not a scale-free network. Its degree distribution is concentrated entirely at degree two, with no power-law tail; degree-preserving edge rewiring cannot change that.

No. A scale-free network is characterized by an approximately power-law tail in its degree distribution, $P(k)\propto k^{-\gamma }$ over a meaningful range. clustering coefficients and mean graph distances do not supply that distribution. One would need the degrees of vertices, enough observations over a substantial range, and a fitted power-law model assessed against alternatives. Four vertices cannot establish such a tail, even if their degrees were supplied. small-worldness alone does not imply a scale-free network.

network resilience depends on which function must persist, which disturbances are applied and how the network responds over time. Local interaction motifs, global architecture and physical operating constraints can protect different properties, often with tradeoffs. A graph that remains connected is not necessarily a functioning regulatory system, and a system that changes its output during a stimulus has not necessarily failed.

It helps to distinguish network robustness, meaning persistence of a specified property under perturbation, from network resilience, which also considers recovery after a disturbance. Relevant performance measures include the maintained output, transient deviation, recovery time, surviving connected component, transport capacity and cost. Disturbances can be brief or persistent, small or destructive, random or targeted, independent or correlated. Consequently, resilience is not one scalar attribute that can be read off a degree distribution.

At the local scale, negative feedback can oppose fluctuations, while network redundancy can provide an alternative route when one component fails. A feed-forward loop can filter stimuli by duration, and positive feedback can stabilize a switched state through bistability. These roles are conditional: a switch that remembers a transient signal can be useful in a developmental decision but undesirable in a homeostatic controller. feedback delay or excessive gain can destabilize a corrective loop; redundancy does little against a disturbance shared by every supposedly independent backup. One must analyze the signs, strengths, kinetics and operating ranges of the interactions, not just count motifs.

Bacterial chemotaxis supplies a concrete cellular example. In Escherichia coli, an attractant change affects receptor-associated kinase activity and hence downstream phosphorylation controlling swimming behavior. Slower receptor methylation and demethylation provide a feedback memory of the stimulus. In an appropriate operating range, the output responds transiently but approximately returns to its earlier level even while the new stimulus persists: robust perfect adaptation. This preserves sensitivity to subsequent changes rather than holding the entire molecular state fixed. The control interpretation identifies conditions under which adaptation does not require precise tuning of every reaction constant.^[ref]

The essential logic can be isolated without pretending to reproduce every reaction. If $y$ is the output and $z$ an internal controller state, supposeWhenever the closed system reaches a stable steady state with an unsaturated integrator, $\dot{z}=0$ forces $y=y_{\ast }$. The controller accumulates the error, so a constant disturbance need not leave a persistent output offset. This is integral action. The equation demonstrates the mechanism, not an unconditional guarantee: the surrounding feedback must be stable, the required control action must be attainable, and the effective set point must remain meaningful under the parameter changes considered.

The example also clarifies what is and is not robust. Returning kinase activity to a set point does not imply an unchanged response time, noise level or swimming trajectory. Strong or sustained stimuli can exhaust the receptor's modification range; deleting an essential feedback component can remove adaptation; correlated changes in controller parameters can shift its set point. Cellular resilience therefore combines biochemical organization with finite resource and timescale constraints. A useful experiment measures the response and recovery after stimuli of different amplitudes, while independently perturbing feedback components, rather than inferring perfect adaptation from one endpoint measurement.

At the global scale, several architectural features matter. Alternative paths and disjoint routes reduce dependence on one component. network modularity can confine damage, although inter-module bridges may then become critical bottlenecks. Short paths improve communication but may also permit rapid spread of a disturbance. Highly connected hubs can support efficient coordination while concentrating vulnerability. Directed links, edge weights and capacities change which routes are usable; a path in an unweighted undirected projection need not carry a signal or the required load in the real system.

The Internet illustrates these distinctions for a real-world network. Nodes can represent routers or autonomous systems, with edges representing different kinds of connectivity; conclusions should state the chosen resolution. In a heterogeneous network, random deletion is likely to hit one of many low-degree nodes, whereas targeted deletion of high-degree nodes can disrupt a large fraction of the remaining paths. The classic attack-tolerance study demonstrated this contrast in model networks and mapped communication networks.^[ref] Its lesson is a vulnerability conditional on the topology and attack rule, not that every network with an apparent power-law tail is invulnerable to arbitrary failures.

Actual Internet resilience also uses routing protocols, spare capacity, replicated services and operational recovery. Those are dynamical and engineering mechanisms additional to the static graph. Removing a node may reroute traffic through surviving links and overload them; failures can then cascade even while many topological paths remain. Conversely, route recomputation may restore service after an initially severe disruption. cascading failure therefore requires a load and dependency model, while geographically correlated failures require an account of shared physical infrastructure. Maintaining connectivity, throughput and low latency are different objectives.

The cellular and communication examples share a design principle but not a single universal architecture. A local controller can reject a persistent input error; a global alternative route can bypass a failed connection. Neither guarantees protection against every perturbation, and both incur costs in energy, components or spare capacity. A sound comparison specifies the disturbance ensemble, the measured function, the recovery interval and the intervention used to test the mechanism. Resilience is produced by matched topology and dynamics, and protection against one class of stimulus can create vulnerability to another.

A scale-free network has a degree distribution with an approximately power-law tail over a meaningful range. Evidence requires degrees from enough vertices to assess that range and compare alternative tail models. Large clustering coefficients or short graph distances alone do not establish this property.

A small-world network combines elevated local clustering coefficients with mean graph distance comparable to a chosen reference ensemble of random graphs. Both the reference ensemble and the clustering convention must be specified. A network can have these properties without having a power-law degree distribution.