Topological ManifoldsMarzio De Corato
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Quantum Lab

Physics, information, quantum computing

From physical phenomena to qubits, circuits, algorithms and protocols: choose a path and explore how representations change as you adjust the model parameters.

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GeoStat Lab

Statistics and R coding for Earth sciences

From samples to landscapes, through data, models and R code: choose a path to compare scenarios, interpret the results and reconstruct the calculation behind them.

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Two independent laboratories, connected by the same curiosity. Return to the course materials whenever you want to revisit a definition, a derivation or an assumption.

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Graph theory and cyber-attacks

Unaffected nodeCompromised node

When devices, services and identities are described through their relationships, system security can be studied as a structural problem in which both a vulnerability and the paths through which it could expose other resources matter.

From connections to attack paths

A graph G = (V, E) consists of vertices V and edges E. In an asset network, vertices may represent devices or services, while edges describe connections, dependencies or trust relationships. An attack graph instead represents conditions, actions or system states, depending on the model, to examine how several steps can combine into a compromise.

G = (V, E)

The analogy with an epidemic helps illustrate how a compromise can spread between nodes, provided that the conditions for each transition are considered. Credentials, permissions, vulnerabilities and configurations determine which paths are feasible; a connection alone describes a possibility of communication.

How this supports defence

Path analysis helps identify the resources reachable from an initial position and the connections that form critical passages. Comparing the graph before and after a change shows how network segmentation, reduced privileges or a vulnerability fix can interrupt particular sequences. A node’s importance depends on its role and context, so its number of connections gives only a partial account of risk.

Reading this figure

Red nodes marked with a cross represent compromised systems in the scenario; green nodes remain unaffected, while pale pulses follow the propagation paths. The map provides a conventional geographical layout for an illustrative network whose links and timing are constructed to clarify the phenomenon. It does not show observed attacks or estimate compromise probabilities.

References

Sheyner et al. — Automated Generation and Analysis of Attack Graphs, Carnegie Mellon University

NIST IR 7788 — Security Risk Analysis of Enterprise Networks Using Probabilistic Attack Graphs

Natural Earth cartography — public-domain data

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Topological Manifolds

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